Sometimes the answer is both. For example, if you have a relatively long name, you want to have an easy way to allow mobile users to type in that name with a short acronym. Example: General Motors (not very long) and you have the domain GeneralMotors.com So that is your brand. You also have GM.com (very short) which allows your actual and potential clients to get to you easily, no matter what the bias of a given search engine may be or become. As the actual brand becomes more difficult to spell, the need for a short alternative domain brand grows exponentially.
Hope the example was useful.